How to Sync iMessage to Mac

Step1: On your iPhone, to go Settings > Messages > Send & Receive.   You’ll need the information shown on this screen for your Mac settings.

Step2: On your Mac, open the Messages app. Click the Messages menu.

Step3: Click Preferences. And Click the iMessage tab.

Confirm the Apple Account you’re logged into here is the same as the one you use on your iPhone. If not, click Sign Out and then sign in with that Apple Account. Check all the boxes in the You can be reached for messages at: section that are checked on your iPhone (see Step 1 for that information). This way, when someone texts you—to your phone number or any email address you may use with iMessage—they will arrive on both devices.  Match the Start new conversations from: drop-down on your Mac to the same setting on the iPhone. It ensures that any new message you send will be attached to the same phone number or email address on both devices and stay in a single message thread.

Credit : https://www.lifewire.com/sync-imessage-to-mac-5211916